Understanding the Domain of 10/ln x for Confused Learners

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The domain of the expression 10/ln(x) is determined by the conditions under which ln(x) is defined and non-zero. The natural logarithm, ln(x), is defined for x in the interval (0, ∞). It is crucial to note that ln(x) equals zero when x equals 1, which makes the expression 10/ln(x) undefined at that point. Therefore, the domain of 10/ln(x) is (0, 1) ∪ (1, ∞). Understanding these constraints is essential for correctly identifying the domain of the function.
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I'm confused on this domain question

10/ln x

I really don't know where to start. Do I graph it and then go from there? :confused:
 
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what's the domain of ln(x)?
 
0, infinity
 
Basically I take it you are looking for the set of all x values for which the expression 10/ln(x) is defined, correct?

So think to yourself, when is that expression not defined?

(When you answered "0,infinity", did you mean (0,infinity)?)
 
well if it equals -10, then I think that would make the numerator zero.

But I think I need to look at the denominator. So it cannot be zero
 
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Rusho said:
well if it equals -10, then I think that would make the numerator zero.

But I think I need to look at the denominator. So it cannot be zero

Yes, that's true. For what value of x is ln x= 0?

Also, the domain of ln x is the open interval (0, \infty).
Your "0, infinity" is ambiguous. 0 is not in the domain of ln x.
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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